Problem: Simplify the following expression: $ r = \dfrac{-10}{9} - \dfrac{z - 10}{z - 2} $
Explanation: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{z - 2}{z - 2}$ $ \dfrac{-10}{9} \times \dfrac{z - 2}{z - 2} = \dfrac{-10z + 20}{9z - 18} $ Multiply the second expression by $\dfrac{9}{9}$ $ \dfrac{z - 10}{z - 2} \times \dfrac{9}{9} = \dfrac{9z - 90}{9z - 18} $ Therefore $ r = \dfrac{-10z + 20}{9z - 18} - \dfrac{9z - 90}{9z - 18} $ Now the expressions have the same denominator we can simply subtract the numerators: $r = \dfrac{-10z + 20 - (9z - 90) }{9z - 18} $ Distribute the negative sign: $r = \dfrac{-10z + 20 - 9z + 90}{9z - 18}$ $r = \dfrac{-19z + 110}{9z - 18}$